Monotonicity and sharp inequalities related to complete (p, q)-elliptic integrals of the first kind

With the aid of the monotone L’Hôpital rule, the authors verify monotonicity of some functions involving complete (p, q)-elliptic integrals of the first kind and the inverse of generalized hyperbolic tangent function, derive several sharp inequalities of complete (p, q)-elliptic integrals of the first kind, and generalize some known sharp approximation of complete elliptic integrals of the first kind. 2020 Mathematics Subject Classification. 33E05, 33C75. Funding. The first author was partially supported by the Project for Combination of Education and Research Training at Zhejiang Institute of Mechanical and Electrical Engineering under Grant No. S027120206. Manuscript received 17th March 2020, revised 15th September 2020 and 13th October 2020, accepted 14th October 2020. ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 962 Fei Wang and Feng Qi 1. Motivation and main results For z ∈C and n ∈ {0}∪N, the rising factorial (z)n is defined [17] by (z)n = n−1 ∏ `=0 (z +`) = { z(z +1) · · · (z +n −1), n ≥ 1; 1, n = 0. It can also be called the Pochhammer symbol or shifted factorial. The hypergeometric function F (a,b;c; z) is defined [23, p. 108, (5.3)] by F (a,b;c; z) = ∞ ∑ n=0 (a)n(b)n (c)n zn n! , |z| < 1 (1) for a,b,c ∈C with c 6= 0,−1,−2, . . . . The complete elliptic integrals of the first and second kinds K (r ) and E (r ) can be expressed by K (r ) = ∫ π/2 0 dφ √ 1− r 2 sin2φ = π 2 F ( 1 2 , 1 2 ;1;r 2 )


Motivation and main results
For z ∈ C and n ∈ {0} ∪ N, the rising factorial (z) n is defined [17] by It can also be called the Pochhammer symbol or shifted factorial.The hypergeometric function F (a, b; c; z) is defined [23, p. 108, (5.3)] by for a, b, c ∈ C with c = 0, −1, −2, . . . .
The complete (p, q)-elliptic integrals of the first and second kinds were defined in [12,22] by K p,q (r ) = π p,q /2 0 1 − r q sin q p,q t 1/p−1 dt and E p,q (r ) complete elliptic integrals of the first and second kinds.For p, q ∈ (1, ∞) and r ∈ [0, 1), the complete (p, q)-elliptic integrals of the first and second kinds can be represented [11,12,22] in terms of the hypergeometric functions F (a, b; c; z) by and C. R. Mathématique, 2020, 358, n 8, 961-970 In [3], the double inequality was obtained, where arctanh r denotes the inverse of hyperbolic tangent function.In [20], the double inequality was proved to be valid if and only if α ≤ 1 − π 2 and β ≥ 0. In [15] and [16,Section 9], among other things, the inequalities π arcsin r 2r and were derived from the Čebyšev integral inequality [18].In [1], the double inequality was sharpened by α 1 = 3 4 and β 1 = 1.In [7], among other things, it was obtained that π In [35], the double inequalities were established.In [24], we discussed monotonicity and some inequalities related to complete elliptic integrals of the second kind E (r ).
We observe that (1) because arctanh r = 1 2 ln 1+r 1−r , the upper bounds in (4) and ( 6) and the best possible bounds in (5) and ( 8) are the same one which cannot compare with the upper bound in (11) on (0, 1); (2) the lower bound in (8) is clearly better than the corresponding one in (4), the lower bounds in ( 5) and ( 6) cannot compare with each other on (0, 1), the lower bounds in ( 5) and ( 8) cannot compare with each other on (0, 1), the lower bound in (8) is better than the corresponding one in (6), and the lower bound in (11) cannot compare with the corresponding ones in (4) to ( 8); (3) the lower bound in (10) is better than the corresponding one in (9) and the upper bounds in (10) and ( 9) cannot compare with each other on (0, 1).
These observations can be verified by plotting via the Wolfram Mathematica 11.1.
In this paper, with the aid of the monotone L'Hôpital rule, we will use a new and concise method to generalize the above inequalities and monotonicity for functions involving K (r ) and E (r ) to those involving complete (p, q)-elliptic integrals K p,q (r ) and E p,q (r ), to reveal monotonicity of several functions involving K p,q (r ), E p,q (r ), and the inverse of generalized hyperbolic tangent function, and to improve inequalities ( 4), ( 5), (8), and (12).
Our main results can be stated as the following theorems.

Lemmas and their proofs
For proving our main results, we need the following known results and lemmas.
The double inequality (21) follows from monotonicity of f (r ).The proof of Lemma 7 is complete.

Proofs of main results
Now we start out to prove our main results.
Proof of Theorem 1.By (18), direct differentiating F (r ) gives By Lemma 7, we have The double inequality ( 14) follows from monotonicity of F (r ).The proof of Theorem 1 is complete.
By the L'Hôpital rule, it follows that and Thus, the double inequality (17) holds.The proof of Theorem 3 is complete.
Remark 13.When p = q = 2, the inequality (20) becomes 1 − r 2 < E (r ) K (r ) < 1 − r 2 2 whose upper bound is worse than (7).This means that the inequality ( 7) is much better.Remark 14. Interested readers who are curious about this paper not only just want to know the main research content of this paper, but also want to know the research background and research progress in this field.Enriching references as many as possible is very important for these readers.So we list several recently published papers [5,8,9,19,[26][27][28][29][30][31][32][33][34], which are closely related to the topic of this paper, to the list of references of this paper.

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).For a, b ∈ R with a < b, let f and g be continuous on [a, b], differentiable on (a, b), and g = 0 on (a, b).If the ratio f g is increasing on (a, b), then both of the functions f (x)− f (a) g (x)−g (a)